Theorem crnggrpd 20265

Description: A commutative ring is a group. (Contributed by SN, 16-May-2024.)

Hypothesis

Ref	Expression
crngringd.1	⊢ (𝜑 → 𝑅 ∈ CRing)

Assertion

Ref	Expression
crnggrpd	⊢ (𝜑 → 𝑅 ∈ Grp)

Proof of Theorem crnggrpd

Step	Hyp	Ref	Expression
1		crngringd.1	. . 3 ⊢ (𝜑 → 𝑅 ∈ CRing)
2	1	crngringd 20264	. 2 ⊢ (𝜑 → 𝑅 ∈ Ring)
3	2	ringgrpd 20260	1 ⊢ (𝜑 → 𝑅 ∈ Grp)

Syntax hints:   → wi 4   ∈ wcel 2106  Grpcgrp 18964  CRingccrg 20252

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-ring 20253  df-cring 20254

This theorem is referenced by:  fermltlchr  21562  psdmul  22188  psd1  22189  ply1chr  22326  ply1fermltlchr  22332  erlbr2d  33251  rlocaddval  33255  rloccring  33257  rloc0g  33258  rlocf1  33260  fracerl  33288  evl1deg1  33581  evl1deg2  33582  evl1deg3  33583  ply1dg1rt  33584  irngss  33702  irredminply  33722  algextdeglem4  33726  algextdeglem5  33727  aks6d1c1p3  42092  aks6d1c2lem4  42109  aks6d1c6lem2  42153  aks5lem2  42169  evlsbagval  42553  selvvvval  42572  evlselv  42574  selvadd  42575  prjcrv0  42620